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The Z-score is a statistical measurement that describes a data point's relation to the mean of a group of data points. It is measured in terms of standard deviations from the mean. A Z-score can help identify how unusual or typical a certain value is within a distribution. If you have a Z-score of zero, it means that the data point's score is identical to the mean score. The Z-score formula is given by:

• \[ Z = \frac{X - \mu}{\sigma} \] where:
  • \(Z\) is the Z-score,
  • \(X\) is the value of the data point,
  • \(\mu\) is the mean of the data set,
  • \(\sigma\) is the standard deviation of the data set.

To find the Z-score for a particular percentile, like in our exercise, you need to find the point in your data set that is higher than the percentage of other points. In other words, the 90th percentile in our example means we're looking for a Z-score where only 10% of the data is higher than this point. The value of the Z-score corresponding to the 90th percentile is approximately 1.28.

A percentile is a measure used in statistics indicating the value below which a given percentage of observations might fall. It helps in understanding the relative standing of a particular value within a data set. For example, if you score in the 90th percentile on a test, this means you scored better than 90% of test-takers.
Percentiles are extremely useful in comparing values from different data sets, making them an important tool in statistics for understanding data distribution.

• The 90th percentile of a dataset is the value below which 90% of the observations fall.
• It can be similarly defined for other percentages.
• Percentiles offer a broader view of the position of a score in the dataset as compared to just the average or median.

In our exercise, we are interested in the 90th percentile, because it tells us the level of gas mileage that surpasses 90% of the vehicles. This is associated with the 'top 10%' threshold as given in the problem.

Mean and standard deviation are two fundamental concepts in statistics that summarize important aspects of a data set. The **mean**, or average, is simply the sum of all values divided by their number. It provides the central point of a data set. For mileage in cars, this might represent the average miles per gallon (mpg) results for the dataset.

• Mean (\(\mu\)) = \( \frac{\text{Sum of all data points}}{\text{Number of data points}} \)

**Standard deviation**, on the other hand, measures the dispersion or spread of data around the mean. If the data points are close to the mean, the standard deviation will be smaller. However, a higher standard deviation indicates a wider range of values. For gas mileage, this could show how much individual vehicles vary in their efficiency.

• Standard Deviation (\(\sigma\)) = \( \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \) where \(X_i\) are the values and \(N\) is the number of values.

In our exercise, the mean is 18.7 mpg and the standard deviation is 4.3 mpg. Recognizing these terms helps us understand the distribution and variability of gas mileages within the 1120 vehicles studied, allowing us to calculate the Z-score and hence find the gas mileage that fits within a defined percentile.